Issue 35

E. Dall’Asta et alii, Frattura ed Integrità Strutturale, 35 (2016) 161-171; DOI: 10.3221/IGF-ESIS.35.19

algorithm choses the disparity solution with the minimum cost, using algorithms like Dynamic Programming (DP) [18], Graph Cuts [19] or Belief Propagation [20]. In a DP framework, as the ones implemented in many SGM software packages [21] the cost   L , d  r p of the pixel located in p at disparity d , along the path direction r is defined recursively as:           i i i L , d C , d min L , d Δd P Δd       r r p p p r (6) where the first term is the similarity cost associated with a disparity value of d, whereas the second term evaluates the regularity of the disparity field, adding a penalty term P , function of disparity changes i (Δd ) with respect to the previous point in the considered matching path r . The final aggregated cost, that takes into account all the different r paths, is defined as     r r S , d L , d   p p (7) and, for sub-pixel estimation of the final disparity solution, the position of the minimum is usually calculated by fitting a quadratic curve through the cost values of neighboring pixels. Indeed, these methods successfully combine concepts of global and local stereo methods for accurate, pixel-wise matching at low runtime and, consequently, works particularly well for 1D displacement (or disparity) field calculation. In many applications, e.g. in stereo-vision problems, the images to be matched can be transformed so that the displacement between conjugated corresponding points occurs always along the same direction (e.g. see rectified images in [22]). In DIC applications (as well as other applications like Particle Image Velocimetry, PIV [23]) the displacements of conjugated points are actually two-dimensional and traditional SGM algorithms cannot be used. In particular eq. (8) should be extended to consider the 2D search domain:           i,j j i i j L , d C , dx, dy min L , dx Δd , dy Δd P Δd , Δd        r r p p p r (8) for eq. (7)) along the principal image plane axis. The high amount of memory and calculations required by the original SGM further increases introducing the 2D disparity search option. It is easy to note that a 1D disparity search domain makes the complexity of the problem proportional to   O m n d   , where m and n are the pixel resolution of the image and d is the disparity search range, while, in the novel approach, the complexity tends to   2 O m n d   . For this reason, a multi-resolution scheme is implemented to limit the disparity search range as much as possible. At the same time, being the total disparity range usually very wide, especially with highly deformable materials, a data structure that saves the expected central displacement value of each pixel is used to further limit the disparity range. The introduction of regularity constraints allows the use of very small templates (usually 1 to 5 pixels) making the method particularly robust where shape discontinuities are present as well as when high deformations occur. Even with a very simple cost function for the similarity term C(p,dx,dy), e.g. the Sum of Absolute Difference (SAD) [24], where no patch deformation are considered (i.e. the patch can only translate), the method can produce good results, at least if image deformation in less than 30-35%. In all these cases the most important tuning is represented by the penalty term P which involves the correct identification and application of a parametric regularization function applied to displacement differences of adjacent pixels. General precaution in DIC application to highly deformable materials Both the previously described image matching techniques can successfully applied to highly deformable materials. Nonetheless some precaution should be provided to ensure that the final results is as accurate and affordable as possible. First of all, sample preparation must be designed so that every area of the specimen is uniformly contrasted and no monochromatic area are present, since such areas, at high deformation rates, can become much bigger and can produce unpredictable results, regardless of the matching algorithm used. Where dx and dy , and i Δd and j Δd , are respectively the displacement components and the disparity changes (as defined

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